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17th-octave temperaments: Difference between revisions

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17edo is a "wheel" for some fractional-octave temperaments. The most notable relationship is the tempering out of [[septendecima]], the amount by which seventeen [[25/24]] chromatic semitones exceed an octave.
{{Technical data page}}
==Chlorine==
{{Infobox fractional-octave|17}}
See [[Ragismic microtemperaments#Chlorine]]
[[17edo]] is a "wheel" for some [[fractional-octave temperaments]]. The most notable relationship is the tempering out of the [[septendecima]], the amount by which seventeen [[25/24]] chromatic semitones exceed an octave.
==Leaves==
Defined as the 323 & 2023 temperament. 2 generators reach [[17/13]], 7 generators reach [[5/4]], 10 generators produce [[13/11]].


== Gothic ==
The gothic temperament is associated with the [[17-comma]]. It used to be known as '''septendecic'''.
[[Subgroup]]: 2.3.5
[[Comma list]]: 134217728/129140163
{{mapping|legend=1| 17 27 0 | 0 0 1 }}
: mapping generators: ~256/243, ~5
[[Optimal tuning]]s:
* [[CTE]]: ~256/243 = 70.5882{{c}} (1\17), ~5/4 = 386.3137{{c}} (~20480/19683 = 33.3725{{c}})
* [[CWE]]: ~256/243 = 70.5882{{c}} (1\17), ~5/4 = 388.2316{{c}} (~20480/19683 = 35.2904{{c}})
{{Optimal ET sequence|legend=1| 17c, 34, 323bbcc, 357bbcc, 391bbcc }}
[[Badness]] (Sintel): 12.7
=== 7-limit ===
Gothic is identical to 17et in the no-5 subgroups, but has an independent generator for prime 5.
[[Subgroup]]: 2.3.5.7
[[Comma list]]: 64/63, 17496/16807
{{Mapping|legend=1| 17 27 0 48 | 0 0 1 0 }}
[[Optimal tuning]]s:
* [[CTE]]: ~28/27 = 70.588{{c}} (1\17), ~5/4 = 386.314{{c}}
* [[CWE]]: ~28/27 = 70.588{{c}} (1\17), ~5/4 = 391.765{{c}}
{{Optimal ET sequence|legend=1| 17c, 34d }}
[[Badness]] (Sintel): 3.44
=== 11-limit ===
Subgroup: 2.3.5.7.11
Comma list: 64/63, 99/98, 243/242
Mapping: {{Mapping| 17 27 0 48 59 | 0 0 1 0 0 }}
Optimal tunings:
* CTE: ~28/27 = 70.588{{c}} (1\17), ~5/4 = 386.314{{c}}
* CWE: ~28/27 = 70.588{{c}} (1\17), ~5/4 = 392.472{{c}}
{{Optimal ET sequence|legend=0| 17c, 34d }}
Badness (Sintel): 1.77
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Comma list: 64/63, 78/77, 99/98, 144/143
Mapping: {{Mapping| 17 27 0 48 59 63 | 0 0 1 0 0 0 }}
Optimal tunings:
* CTE: ~27/26 = 70.588{{c}} (1\17), ~5/4 = 386.314{{c}}
* CWE: ~27/26 = 70.588{{c}} (1\17), ~5/4 = 392.129{{c}}
{{Optimal ET sequence|legend=0| 17c, 34d }}
Badness (Sintel): 1.35
== Chlorine ==
The name of chlorine temperament comes from chlorine, the 17th element, and has no relation to the [[chlorisma]].
Chlorine temperament has a period of 1/17 octave. It tempers out the [[septendecima]], {{monzo| -52 -17 34 }}, by which 17 chromatic semitones (25/24) exceed an octave. This temperament can be described as 289 & 323 temperament, which tempers out {{monzo| -49 4 22 -3 }} as well as the ragisma. Not only the semitwelfth, but also the ~5/4 can be used as a generator.
[[Subgroup]]: 2.3.5
[[Comma list]]: {{monzo| -52 -17 34 }}
{{Mapping|legend=1| 17 0 26 | 0 2 1 }}
: mapping generators: ~25/24, ~{{monzo| 26 9 -17 }}
[[Optimal tuning]]s:
* [[CTE]]: ~25/24 = 70.588{{c}} (1\17), ~{{monzo| 26 9 -17 }} = 950.982{{c}}
* [[CWE]]: ~25/24 = 70.588{{c}} (1\17), ~{{monzo| 26 9 -17 }} = 950.978{{c}}
{{Optimal ET sequence|legend=1| 34, 153, 187, 221, 255, 289, 323, 612, 3349, 3961, 4573, 5185, 5797 }}
[[Badness]] (Sintel): 1.81
=== 7-limit ===
[[Subgroup]]: 2.3.5.7
[[Comma list]]: 4375/4374, {{monzo| -49 4 22 -3 }}
{{Mapping|legend=1| 17 0 26 -87 | 0 2 1 10 }}
[[Optimal tuning]]s:
* [[CTE]]: ~25/24 = 70.588{{c}} (1\17), ~{{monzo| 24 -5 -9 2 }} = 950.998{{c}}
* [[CWE]]: ~25/24 = 70.588{{c}} (1\17), ~{{monzo| 24 -5 -9 2 }} = 950.999{{c}}
{{Optimal ET sequence|legend=1| 289, 323, 612, 935, 1547 }}
[[Badness]] (Sintel): 1.05
=== 11-limit ===
Subgroup: 2.3.5.7.11
Comma list: 4375/4374, 41503/41472, 1879453125/1879048192
Mapping: {{mapping| 17 0 26 -87 207 | 0 2 1 10 -11 }}
Optimal tunings:
* CTE: ~25/24 = 70.588{{c}} (1\17), ~693/400 = 950.978{{c}}
* CWE: ~25/24 = 70.588{{c}} (1\17), ~693/400 = 950.975{{c}}
{{Optimal ET sequence|legend=0| 289, 323, 612 }}
Badness (Sintel): 2.11
== Leaves ==
Defined as the {{nowrap|323 & 2023}} temperament. 2 generators reach [[17/13]], 7 generators reach [[5/4]], 10 generators produce [[13/11]].
[[Subgroup]]: 2.3.5.7
[[Comma list]]: 1729951171875/1727094849536, {{monzo|10 39 -14 -14}}
[[Mapping]]: [{{val|17 10 31 9}}, {{val|0 14 7 32}}]
: mapping generators: ~25/24, ~6125/5832
[[Optimal tuning]]s:
* [[CTE]]: ~25/24 = 70.588{{c}} (1\17), ~6125/5832 = 85.427{{c}}
* [[CWE]]: ~25/24 = 70.588{{c}} (1\17), ~6125/5832 = 85.426{{c}}
{{Optimal ET sequence|legend=1| 323, 1700d, 2023, 2346}}
[[Badness]] (Sintel): 32.032
=== 11-limit ===
Subgroup: 2.3.5.7.11
Comma list: 160083/160000, 198359290368/198165034375, 1729951171875/1727094849536
Mapping: [{{val|17 10 31 9 106}}, {{val|0 14 7 32 -39}}]
Optimal tunings:
* CTE: ~25/24 = 70.588{{c}} (1\17), ~6125/5832 = 85.421{{c}}
* CWE: ~25/24 = 70.588{{c}} (1\17), ~6125/5832 = 85.420{{c}}
{{Optimal ET sequence|legend=0| 323, 1700d, 2023, 2346e}}
Badness (Sintel): 20.456
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 160083/160000, 928125/927472, 1990656/1990625, 20726199/20706224
Comma list: 160083/160000, 928125/927472, 1990656/1990625, 4831530/4826809


Mapping: [{{val|17 10 31 9 106 98}}, {{val|0 14 7 32 -39 -29}}]
Mapping: [{{val|17 10 31 9 106 98}}, {{val|0 14 7 32 -39 -29}}]


Mapping generators: ~25/24, ~1024/975
Optimal tunings:
* CTE: ~25/24 = 70.588{{c}} (1\17), ~1024/975 = 85.421{{c}}
* CWE: ~25/24 = 70.588{{c}} (1\17), ~1024/975 = 85.420{{c}}
 
{{Optimal ET sequence|legend=0| 323, 1700d, 2023, 2346e}}
 
Badness (Sintel): 10.096


Optimal tuning (CTE): ~1024/975 = 85.421
=== 17-limit ===
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17
Subgroup: 2.3.5.7.11.13.17


Comma list: 57375/57344, 111537/111475, 140800/140777, 111537/111475, 1026675/1026256
Comma list: 39325/39304, 57375/57344, 71874/71825, 111537/111475, 140800/140777


Mapping: [{{val|17 10 31 9 106 98 107}}, {{val|0 14 7 32 -39 -29 -31}}]
Mapping: [{{val|17 10 31 9 106 98 107}}, {{val|0 14 7 32 -39 -29 -31}}]


Mapping generators: ~25/24, ~765/728
Optimal tunings:
* CTE: ~25/24 = 70.588{{c}} (1\17), ~765/728 = 85.421{{c}}
* CWE: ~25/24 = 70.588{{c}} (1\17), ~765/728 = 85.421{{c}}
 
{{Optimal ET sequence|legend=0| 323, 1700d, 2023, 2346e}}
 
Badness (Sintel): 6.678


Optimal tuning (CTE): ~765/728 = 85.421
{{Navbox fractional-octave}}


Vals: {{EDOs|323, 1700, 2023}}
[[Category:17edo]]
[[Category:17edo]]
[[Category:Temperament collections]]
[[Category:Rank 2]]
Retrieved from "https://en.xen.wiki/w/17th-octave_temperaments"